Limiting behavior of periodic measures of stochastic Schrödinger delay lattice systems

In this paper, we investigate the limiting behavior of periodic measures of stochastic Schrödinger lattice systems with time delay. We first give the existence of periodic measures of the systems, and then show that the set of all periodic measures of the perturbed system is weakly compact. We finally prove that every limit point of periodic measures of the stochastic delay lattice systems must be a periodic measure of the corresponding limiting system as the intensity of noise converges or the time delay approaches zero.